Either way can work. Getting to millions of messages is going to be the hard part, not storing them.
As with all data storage, the question is usually how do you want to access that data. I don't have experience with Postgres, but a lot of (older) experience with MySQL, and MySQL makes a pretty reasonable key-value storage engine, so I'd expect Postgres to do ok at that too.
I'm a big fan of pushing the messages to the clients, so the server is only holding messages in transit. Each client won't typically have millions of messages or even close, so you have freedom to store things how you want there, and the servers have more of a queue per user than a database --- but you can use a RDBMS as a queue if you want, especially if you have more important things to work on.
As with all data storage, the question is usually how do you want to access that data. I don't have experience with Postgres, but a lot of (older) experience with MySQL, and MySQL makes a pretty reasonable key-value storage engine, so I'd expect Postgres to do ok at that too.
I'm a big fan of pushing the messages to the clients, so the server is only holding messages in transit. Each client won't typically have millions of messages or even close, so you have freedom to store things how you want there, and the servers have more of a queue per user than a database --- but you can use a RDBMS as a queue if you want, especially if you have more important things to work on.